In a small bowl or measuring cup, whisk together the vegetable broth, sweet chili sauce, soy sauce, lime juice, and red chili flakes. Set aside.
Using a clean kitchen towel, or paper towels, dry off the tofu block, squeezing it to remove as much excess water as possible. Cut the tofu into 1 inch cubes and sprinkle them lightly with salt. In a large bowl, toss the tofu with the cornstarch until evenly coated.
In a large skillet, heat 2 tablespoons of oil over medium heat. Add the tofu in a single layer, leaving behind any excess cornstarch. Cook the tofu, turning frequently, until browned and crispy on all sides. This takes a good 10-15 minutes. Remove the tofu to a plate.
In the same pan, heat the remaining tablespoon of oil. Add the broccoli florets, season with a pinch of salt and pepper, and cook, stirring frequently, until bright green and crisp-tender, about 5 minutes. Add the garlic and cook until fragrant, 30 seconds more. Transfer broccoli to the bowl with the tofu.
Add the sauce to the pan and let it simmer, and thicken, and reduce slightly for 3-4 minutes.
Add the tofu and broccoli back to the pan and stir to coat and warm up. Garnish with green onions and sesame seeds, and serve over rice.